i - start insert mode at cursor
I - insert at the beginning of the line
a - append after the cursor
A - append at the end of the line
o - open (append) blank line below current line (no need to press return)
O - open blank line above current line
ea - append at end of word
Esc - exit insert mode

Editing

r - replace a single character (does not use insert mode)
J - join line below to the current one
cc - change (replace) an entire line
cw - change (replace) to the end of word
c$ - change (replace) to the end of line
s - delete character at cursor and subsitute text
S - delete line at cursor and substitute text (same as cc)
xp - transpose two letters (delete and paste, technically)
u - undo
. - repeat last command

/pattern - search for pattern
?pattern - search backward for pattern
n - repeat search in same direction
N - repeat search in opposite direction
:%s/old/new/g - replace all old with new throughout file
:%s/old/new/gc - replace all old with new throughout file with confirmations